Assume G is a superstable locally modular group. We describe for any countable model M of Th(G) the quotient group G(M) / Gm(M). Here Gm is the modular part of G. Also, under some additional assumptions we describe G(M) / Gm(M) relative to G⁻(M). We prove Vaught's Conjecture for Th(G) relative to Gm and a finite set provided that ℳ(G) = 1 and the ring of pseudoendomorphisms of G is finite.
References
[1] Buechler, S., "The classification of small weakly minimal sets. I", pp. 32--71 in Classification Theory (Chicago, 1985), edited by J. T. Baldwin, vol. 1292 of Lecture Notes in Mathematics, Springer, Berlin, 1987. MR1033022 0655.03022 10.1007/BFb0082231[1] Buechler, S., "The classification of small weakly minimal sets. I", pp. 32--71 in Classification Theory (Chicago, 1985), edited by J. T. Baldwin, vol. 1292 of Lecture Notes in Mathematics, Springer, Berlin, 1987. MR1033022 0655.03022 10.1007/BFb0082231
[2] Buechler, S., "Vaught's conjecture for superstable theories of finite rank", preprint, 1993. 1175.03019[2] Buechler, S., "Vaught's conjecture for superstable theories of finite rank", preprint, 1993. 1175.03019
[3] Hrushovski, E., "Locally modular regular types", pp. 132--64 in Classification Theory (Chicago, 1985), edited by J. T. Baldwin, vol. 1292 of Lecture Notes in Mathematics, Springer, Berlin, 1987. MR1033027 0643.03024 10.1007/BFb0082236[3] Hrushovski, E., "Locally modular regular types", pp. 132--64 in Classification Theory (Chicago, 1985), edited by J. T. Baldwin, vol. 1292 of Lecture Notes in Mathematics, Springer, Berlin, 1987. MR1033027 0643.03024 10.1007/BFb0082236
[4] Loveys, J., "Abelian groups with modular generic", The Journal of Symbolic Logic, vol. 56 (1991), pp. 250--59. MR1131743 0734.03023 10.2307/2274917 euclid.jsl/1183743564
[4] Loveys, J., "Abelian groups with modular generic", The Journal of Symbolic Logic, vol. 56 (1991), pp. 250--59. MR1131743 0734.03023 10.2307/2274917 euclid.jsl/1183743564
[5] Newelski, L., "On type definable subgroups of a stable group", Notre Dame Journal of Formal Logic, vol. 32 (1991), pp. 173--87. MR1122992 0747.03015 10.1305/ndjfl/1093635743 euclid.ndjfl/1093635743
[5] Newelski, L., "On type definable subgroups of a stable group", Notre Dame Journal of Formal Logic, vol. 32 (1991), pp. 173--87. MR1122992 0747.03015 10.1305/ndjfl/1093635743 euclid.ndjfl/1093635743
[6] Newelski, L., "A model and its subset", The Journal of Symbolic Logic, vol. 57 (1992), pp. 644--58. MR1169199 0774.03015 10.2307/2275297 euclid.jsl/1183743982
[6] Newelski, L., "A model and its subset", The Journal of Symbolic Logic, vol. 57 (1992), pp. 644--58. MR1169199 0774.03015 10.2307/2275297 euclid.jsl/1183743982
[7] Newelski, L., "Scott analysis of pseudotypes", The Journal of Symbolic Logic, vol. 58 (1993), pp. 648--63. MR1233930 0781.03016 10.2307/2275225 euclid.jsl/1183744253
[7] Newelski, L., "Scott analysis of pseudotypes", The Journal of Symbolic Logic, vol. 58 (1993), pp. 648--63. MR1233930 0781.03016 10.2307/2275225 euclid.jsl/1183744253
[8] Newelski, L., "Meager forking", Annals of Pure and Applied Logic, vol. 70 (1994), pp. 141--75. MR1321462 0817.03017 10.1016/0168-0072(94)90028-0[8] Newelski, L., "Meager forking", Annals of Pure and Applied Logic, vol. 70 (1994), pp. 141--75. MR1321462 0817.03017 10.1016/0168-0072(94)90028-0
[9] Newelski, L., "$\mathcalM$-rank and meager types", Fundamenta Mathematicae, vol. 146 (1995), pp. 121--39. MR1314978 0829.03016[9] Newelski, L., "$\mathcalM$-rank and meager types", Fundamenta Mathematicae, vol. 146 (1995), pp. 121--39. MR1314978 0829.03016
[10] Newelski, L., "$\mathcalM$-rank and meager groups", Fundamenta Mathematicae, vol. 150 (1996), pp. 149--71. MR1391297 0865.03022[10] Newelski, L., "$\mathcalM$-rank and meager groups", Fundamenta Mathematicae, vol. 150 (1996), pp. 149--71. MR1391297 0865.03022
[11] Newelski, L., "$\mathcalM$-gap conjecture and m-normal theories", Israel Journal of Mathematics, vol. 106 (1998), pp. 285--311. MR1656901 0914.03040 10.1007/BF02773473[11] Newelski, L., "$\mathcalM$-gap conjecture and m-normal theories", Israel Journal of Mathematics, vol. 106 (1998), pp. 285--311. MR1656901 0914.03040 10.1007/BF02773473
[12] Newelski, L., "Flat Morley sequences", The Journal of Symbolic Logic, vol. 64 (1999), pp. 1261--79. MR1779760 0937.03045 10.2307/2586628 euclid.jsl/1183745879
[12] Newelski, L., "Flat Morley sequences", The Journal of Symbolic Logic, vol. 64 (1999), pp. 1261--79. MR1779760 0937.03045 10.2307/2586628 euclid.jsl/1183745879
[13] Newelski, L., "On the topological stability conjecture", pp. 279--92 in Models and Computability (Proceedings of Logic Colloquium, Leeds, 1997), edited by S. B. Cooper and J. K. Truss, vol. 259 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1999. MR1721171 0941.03033[13] Newelski, L., "On the topological stability conjecture", pp. 279--92 in Models and Computability (Proceedings of Logic Colloquium, Leeds, 1997), edited by S. B. Cooper and J. K. Truss, vol. 259 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1999. MR1721171 0941.03033
[14] Newelski, L., "Vaught's conjecture for some meager groups", Israel Journal of Mathematics, vol. 112 (1999), pp. 271--99. MR1714986 0948.03034 10.1007/BF02773485[14] Newelski, L., "Vaught's conjecture for some meager groups", Israel Journal of Mathematics, vol. 112 (1999), pp. 271--99. MR1714986 0948.03034 10.1007/BF02773485
[16] Pillay, A., Geometric Stability Theory, vol. 32 of Oxford Logic Guides, The Clarendon Press, New York, 1996. MR1429864 0871.03023[16] Pillay, A., Geometric Stability Theory, vol. 32 of Oxford Logic Guides, The Clarendon Press, New York, 1996. MR1429864 0871.03023